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NOON State Generation with Phonons in Acoustic Wave Resonators

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NOON State Generation with Phonons in Acoustic Wave Resonators NOON state generation with phonons in acoustic wave resonators assisted by a nitrogen-vacancy-center ensemble Jiu-Ming Li, Ming Hua∗, and Xue-Qun Yan Department of Applied Physics, School of Physical Science and Technology, Tianjin Polytechnic University, Tianjin 300387, China 11 Since the quality factor of an acoustic wave resonator (AWR) reached 10 , AWRs have been regarded as a good carrier of quantum information. In this paper, we propose a scheme to construct a NOON state with two AWRs assisted by a nitrogen-vacancy-center ensemble (NVE). The two AWRs cross each other vertically, and the NVE is located at the center of the crossing. By considering the decoherence of the system and using resonant interactions between the AWRs and the NVE, and the single-qubit operation of the NVE, a NOON state can be achieved with a fidelity higher than 98.8% when the number of phonons in the AWR is N ≤ 3. Keywords: acoustic wave resonator; phonon; nitrogen-vacancy-center ensemble; quantum entanglement; quantum electrodynamics I. INTRODUCTION Quantum entanglement is not only one of the significant features of quantum mechanics but also an important application in quantum information processing (QIP) [1–14]. Over the past decades, many works have been proposed to creat entanglements using atoms or photons assisted by various quantum systems [15, 16], such as cavity quantum electrodynamics (cavity-QED) [17–23], circuit quantum electrodynamics (circuit-QED) [24–35],trapped ions [36–39], quantum dots [40–45], cold atoms [46, 47], nitrogen-vacancy (NV) centers [48–52], nuclear magnetic resonance [53–57], acoustic wave resonators (AWRs) [58], optomechanical systems [59–62], and atomic ensembles in free space [63]. In recent years, acoustic wave resonators, whose quality-factor (Q-factor) has been increased to 1011 [64], have attracted more and more attention for QIP [65–76]. For example, in 2012, Goryachev et al. [65] measured∼ a quartz bulk acoustic wave resonator and showed that the Q-factor of the resonator can be continually increased to 109. In 2018, Cai et al. [70] proposed a scheme to achieve a single phonon source based on the nonlinearity generated by the four-level NV-centers in a diamond photonic crystal resonator. In 2017, Noguchi et al. [72] demonstrated an ultrasensitive measurement of fluctuations in an AWR and up-converted the excitation in the AWR to an excitation in a microwave resonator in a hybrid system consisting of an AWR, a microwave resonator, and a superconducting qubit. To achieve the QIP with AWRs, many works have studied the coupling between a nonlinear quantum system and an AWR both experimentally and theoretically. Examples include the coupling between an AWR and a superconducting qubit [77–82], an NV-center [68, 83, 84], the phononic QED [67, 85], and the quantum dots [86–88]. Among these couplings, due to the long coherence time [89, 90] and good manipulability of the NV-centers [91–94], the coupling between NV-centers and the AWRs has attracted much attention in recent years. For example, in 2016, Golter et al. [68] realized the strong coupling between a nitrogen-vacancy-center ensemble (NVE) and an AWR, and they also realized the coherent population trapping and optically driven spin transitions. In the same year [69], they demonstrated the quantum control of an NV-center in the resolved-sideband regime by coupling the NV-center to the optical and the AWR fields. In this paper, we propose a simple scheme to generate a NOON state with AWRs in a system consisting of two arXiv:1810.08767v2 [quant-ph] 30 Nov 2018 AWRs coupled to an NVE for the first time. The operations used here are resonant interactions between the AWRs (microwave pulse) and the NVE, which help us to achieve a high fidelity NOON state in a short time. To demonstrate the feasibility of the scheme, we numerically simulate the fidelities of the NOON states with the AWR phonon number as N = 1, 2, and 3 by considering the decoherence of the system, and the fidelities of which reach 99.54%, 99.18%, and 98.80%, respectively. II. GENERATION OF NOON STATE WITH THE AWRS To construct a NOON state with AMRs, we consider a system consisting of two AWRs coupled to an NVE as shown in Fig.1(a). The two AWRs cross each other vertically, and the NVE is located at the center of the crossing. ∗ Corresponding author:[email protected] 2 The electronic ground state of the NV-center has a spin of S = 1 with an energy splitting between the states ms =0 | i and ms = 1 with frequency ω± 2π 2.88 GHz in a zero magnetic field. By applying an external magnetic field | ± i ≈ × B~ parallel to the axis between the nitrogen and the vacancy, the states ms = 1 can be split. For simplicity, the | ± i states of the NV-centers ms =0 , ms = 1 , and ms =1 are labelled as g , e , and u , respectively, with where | i | − i | i | i | i | i the energies are characterized by Eg < Ee and Eg < Eu. ωge (ωgu) is defined as the transition frequency between states g and e ( g and u ). Here, ωge can be tuned by B~ , and ωgu is kept unchanged as 2π 2.88 GHz. The | i | i | i | i × states g and e ( g and u ) of the NVE can be flipped by applying a microwave pulse with frequency ωe (ωu) and | i | i | i | i strength Ωe (Ωu) as shown in Fig.1(b). In addition, the frequencies of the AWRs and the transition g u of the NVE should be far detuned from each other largely. | i ↔ | i FIG. 1: (Color online) (a) Setup of the system consisting of two AWRs coupled to an NVE. (b) Energy levels of an NV-center under an external field B~ . |ms = 0i, |ms = −1i, and |ms = 1i are labelled as |gi, |ei, and |ui, respectively. A microwave pulse with strength Ωe (Ωu) is applied to flip the states |gi and |ei (|gi ↔ |ui) of the NVE. There are two stages with (2N +2M + 3) steps to generate the NOON state. First, we focus on the first stage which contains (2N + 1) steps. In this stage, the two transitions g e and g u of the NVE are set to be far | i ↔ | i | i ↔ | i detuned from AWR2 largely all the time. The initial state of the whole system should be prepared as ψ = g 0 0 . (1) | iI | i | i1 | i2 Here, the subscript 1(2) represents the AWR1(2). The operations of the first stage containing (2N + 1) steps are described as follows: Step 1: A microwave pulse with strength Ωe is applied to resonate with the transition g e of the NVE to form + + | i ↔ | i the Hamiltonian He = ~Ωe σ + H.c. , where σ represents the raising operator of the transition g e . Here, ge ge | i ↔ | i the two transitions of the NVE should be far detuned from AWR1 and AWR2 largely. Then, the state of the system will evolve from ψ to | iI 1 ψ = ( g i e ) 0 0 (2) | i1 √2 | i− | i | i1 | i2 after an operation time t = π/ (4Ωe). Step 2: A microwave pulse with strength Ωu is applied to resonate with the transition g u of the NVE when | i ↔ | i the two transitions of the NVE are detuned from AWR1 and AWR2. After an operation time of t = π/ (2Ωu), state g of the NVE is excited to u with a i phase shift, and the state of the whole system becomes | i | i − i ψ = − ( e + u ) 0 0 . (3) | i2 √2 | i | i | i1 | i2 Step 3: By letting the transition g e of the NVE resonate with AWR (ωge = ω ), state ψ will evolve to | i ↔ | i 1 1 | i2 i ψ = − ( i g 1 + u 0 ) 0 (4) | i3 √2 − | i | i1 | i | i1 | i2 ge ge after an operation time t = π/ (2g1 ). Here, ω1(2) is the frequency of the AWR1(2). g1 is the coupling strength between the AWR and the transition g e of the NVE. 1 | i ↔ | i 3 Step 4: A microwave pulse with strength Ωe is applied to excite the state g to e after an operation time of | i | i t = π/ (2Ωe). Then, the state of the system evolves from ψ to | i3 i ψ = − [ i ( i) e 1 + u 0 ] 0 . (5) | i4 √2 − − | i | i1 | i | i1 | i2 Step 5: As in the step 3, we let the transition g e of the NVE resonate with the AWR1. After an interaction time t = π/ (2gge), the state of the system evolves| i from ↔ | iψ to 1 | i4 i ψ = − [ i ( 1) g 2 + u 0 ] 0 . (6) | i5 √2 − − | i | i1 | i | i1 | i2 Step j (j = 6, 7, ..., 2N + 1): Repeating steps 4 and 5 successively with N + 1 times, the state of the system will evolve to i N−1 ψ 2N+1 = − i ( 1) g N 1 + u 0 1 0 2 . (7) | i √2 h− − | i | i | i | i i | i Then, by tuning the two transitions g e and g u of the NVE to be far detuned from AWR1 largely, we will give the operations of (2M + 2) steps| i ↔ of the| i second| i ↔stage | i to achieve the NOON state as follows: Step 2N + 2: A microwave pulse with strength Ωu is applied to flip the states g and u of the NVE. After an | i | i operation time t = π/ (2Ωu), the state of the system becomes ′ 1 N−1 ψ 1 = − i ( 1) u N 1 0 2 + g 0 1 0 2 .
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